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(C) Total number of students = $4$ girls + $6$ boys = $10$ students.We need to form a group of $4$ students.The total number of ways to select $4$ stu

Vedclass · Accepted Answer

$\frac{5}{42}$

Vedclass · Answer

(C) Total number of students = $4$ girls + $6$ boys = $10$ students.We need to form a group of $4$ students.The total number of ways to select $4$ students from $10$ is $n(S) = {}^{10}C_{4} = \frac{10 	imes 9 	imes 8 	imes 7}{4 	imes 3 	imes 2 	imes 1} = 210$.We want the number of boys to be less than the number of girls in a group of $4$.Possible cases for (boys,girls) such that boys Case $1$: $1$ boy and $3$ girls. Number of ways = ${}^{6}C_{1} 	imes {}^{4}C_{3} = 6 	imes 4 = 24$.Case $2$: $0$ boys and $4$ girls. Number of ways = ${}^{6}C_{0} 	imes {}^{4}C_{4} = 1 	imes 1 = 1$.Total favorable outcomes $n(E) = 24 + 1 = 25$.Probability $P(E) = \frac{n(E)}{n(S)} = \frac{25}{210} = \frac{5}{42}$.

$A$ group of $4$ students is to be formed from among $4$ girls and $6$ boys. What is the probability that the group has less number of boys than the number of girls?

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